## 59.59 Cohomology of a point

As a consequence of the discussion in the preceding sections we obtain the equivalence of étale cohomology of the spectrum of a field with Galois cohomology.

Lemma 59.59.1. Let $S = \mathop{\mathrm{Spec}}(K)$ with $K$ a field. Let $\overline{s}$ be a geometric point of $S$. Let $G = \text{Gal}_{\kappa (s)}$ denote the absolute Galois group. The stalk functor induces an equivalence of categories

$\textit{Ab}(S_{\acute{e}tale}) \longrightarrow \text{Mod}_ G, \quad \mathcal{F} \longmapsto \mathcal{F}_{\overline{s}}.$

Proof. In Theorem 59.56.3 we have seen the equivalence between sheaves of sets and $G$-sets. The current lemma follows formally from this as an abelian sheaf is just a sheaf of sets endowed with a commutative group law, and a $G$-module is just a $G$-set endowed with a commutative group law. $\square$

Lemma 59.59.2. Notation and assumptions as in Lemma 59.59.1. Let $\mathcal{F}$ be an abelian sheaf on $\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}$ which corresponds to the $G$-module $M$. Then

1. in $D(\textit{Ab})$ we have a canonical isomorphism $R\Gamma (S, \mathcal{F}) = R\Gamma _ G(M)$,

2. $H_{\acute{e}tale}^0(S, \mathcal{F}) = M^ G$, and

3. $H_{\acute{e}tale}^ q(S, \mathcal{F}) = H^ q(G, M)$.

Example 59.59.3. Sheaves on $\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}$. Let $G = \text{Gal}(K^{sep}/K)$ be the absolute Galois group of $K$.

1. The constant sheaf $\underline{\mathbf{Z}/n\mathbf{Z}}$ corresponds to the module $\mathbf{Z}/n\mathbf{Z}$ with trivial $G$-action,

2. the sheaf $\mathbf{G}_ m|_{\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}}$ corresponds to $(K^{sep})^*$ with its $G$-action,

3. the sheaf $\mathbf{G}_ a|_{\mathop{\mathrm{Spec}}(K^{sep})}$ corresponds to $(K^{sep}, +)$ with its $G$-action, and

4. the sheaf $\mu _ n|_{\mathop{\mathrm{Spec}}(K^{sep})}$ corresponds to $\mu _ n(K^{sep})$ with its $G$-action.

By Remark 59.23.4 and Theorem 59.24.1 we have the following identifications for cohomology groups:

\begin{align*} H_{\acute{e}tale}^0(S_{\acute{e}tale}, \mathbf{G}_ m) & = \Gamma (S, \mathcal{O}_ S^*) \\ H_{\acute{e}tale}^1(S_{\acute{e}tale}, \mathbf{G}_ m) & = H_{Zar}^1(S, \mathcal{O}_ S^*) = \mathop{\mathrm{Pic}}\nolimits (S) \\ H_{\acute{e}tale}^ i(S_{\acute{e}tale}, \mathbf{G}_ a) & = H_{Zar}^ i(S, \mathcal{O}_ S) \end{align*}

Also, for any quasi-coherent sheaf $\mathcal{F}$ on $S_{\acute{e}tale}$ we have

$H^ i(S_{\acute{e}tale}, \mathcal{F}) = H_{Zar}^ i(S, \mathcal{F}),$

see Theorem 59.22.4. In particular, this gives the following sequence of equalities

$0 = \mathop{\mathrm{Pic}}\nolimits (\mathop{\mathrm{Spec}}(K)) = H_{\acute{e}tale}^1(\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}, \mathbf{G}_ m) = H^1(G, (K^{sep})^*)$

which is none other than Hilbert's 90 theorem. Similarly, for $i \geq 1$,

$0 = H^ i(\mathop{\mathrm{Spec}}(K), \mathcal{O}) = H_{\acute{e}tale}^ i(\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}, \mathbf{G}_ a) = H^ i(G, K^{sep})$

where the $K^{sep}$ indicates $K^{sep}$ as a Galois module with addition as group law. In this way we may consider the work we have done so far as a complicated way of computing Galois cohomology groups.

The following result is a curiosity and should be skipped on a first reading.

Lemma 59.59.4. Let $R$ be a local ring of dimension $0$. Let $S = \mathop{\mathrm{Spec}}(R)$. Then every $\mathcal{O}_ S$-module on $S_{\acute{e}tale}$ is quasi-coherent.

Proof. Let $\mathcal{F}$ be an $\mathcal{O}_ S$-module on $S_{\acute{e}tale}$. We have to show that $\mathcal{F}$ is determined by the $R$-module $M = \Gamma (S, \mathcal{F})$. More precisely, if $\pi : X \to S$ is étale we have to show that $\Gamma (X, \mathcal{F}) = \Gamma (X, \pi ^*\widetilde{M})$.

Let $\mathfrak m \subset R$ be the maximal ideal and let $\kappa$ be the residue field. By Algebra, Lemma 10.153.10 the local ring $R$ is henselian. If $X \to S$ is étale, then the underlying topological space of $X$ is discrete by Morphisms, Lemma 29.36.7 and hence $X$ is a disjoint union of affine schemes each having one point. Moreover, if $X = \mathop{\mathrm{Spec}}(A)$ is affine and has one point, then $R \to A$ is finite étale by Algebra, Lemma 10.153.5. We have to show that $\Gamma (X, \mathcal{F}) = M \otimes _ R A$ in this case.

The functor $A \mapsto A/\mathfrak m A$ defines an equivalence of the category of finite étale $R$-algebras with the category of finite separable $\kappa$-algebras by Algebra, Lemma 10.153.7. Let us first consider the case where $A/\mathfrak m A$ is a Galois extension of $\kappa$ with Galois group $G$. For each $\sigma \in G$ let $\sigma : A \to A$ denote the corresponding automorphism of $A$ over $R$. Let $N = \Gamma (X, \mathcal{F})$. Then $\mathop{\mathrm{Spec}}(\sigma ) : X \to X$ is an automorphism over $S$ and hence pullback by this defines a map $\sigma : N \to N$ which is a $\sigma$-linear map: $\sigma (an) = \sigma (a) \sigma (n)$ for $a \in A$ and $n \in N$. We will apply Galois descent to the quasi-coherent module $\widetilde{N}$ on $X$ endowed with the isomorphisms coming from the action on $\sigma$ on $N$. See Descent, Lemma 35.6.2. This lemma tells us there is an isomorphism $N = N^ G \otimes _ R A$. On the other hand, it is clear that $N^ G = M$ by the sheaf property for $\mathcal{F}$. Thus the required isomorphism holds.

The general case (with $A$ local and finite étale over $R$) is deduced from the Galois case as follows. Choose $A \to B$ finite étale such that $B$ is local with residue field Galois over $\kappa$. Let $G = \text{Aut}(B/R) = \text{Gal}(\kappa _ B/\kappa )$. Let $H \subset G$ be the Galois group corresponding to the Galois extension $\kappa _ B/\kappa _ A$. Then as above one shows that $\Gamma (X, \mathcal{F}) = \Gamma (\mathop{\mathrm{Spec}}(B), \mathcal{F})^ H$. By the result for Galois extensions (used twice) we get

$\Gamma (X, \mathcal{F}) = (M \otimes _ R B)^ H = M \otimes _ R A$

as desired. $\square$

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